Correlated disorder as a way towards robust superconductivity

The study addresses how spatial correlations in nonmagnetic disorder affect superconductivity, particularly in low-dimensional systems where disorder can both suppress and enhance superconducting properties. Classic theory (Abrikosov–Gor'kov; Anderson) predicts s-wave superconductivity is insensitive to weak nonmagnetic disorder, yet experiments show suppression at strong disorder and even superconductor–insulator transitions. In quasi-2D materials, disorder and reduced dimensionality amplify interaction effects, localization, and phase fluctuations, typically reducing superconductivity. Conversely, disorder can enhance local pairing and induce multifractality, raising the gap and potentially Tc. Real materials exhibit correlated disorder (impurities/defects are not fully random), introducing a new length scale competing with BCS coherence length and Fermi wavelength. The research question: Do long-range spatial correlations in the disorder potential make superconductivity more robust, and how do they modify local and global superconducting properties? The purpose is to quantify these effects at zero temperature by solving microscopic BdG equations in 2D with a power-law correlated disorder model and to identify regimes where correlations counteract the detrimental effects of disorder.

Prior theory and experiments show that while weak nonmagnetic disorder leaves s-wave superconductivity largely unaffected, strong disorder suppresses superconductivity, enhances spatial inhomogeneity, increases the gap-to-Tc ratio, and can induce metal-insulator transitions and pseudogaps (Refs. 3–13). Low-dimensional (2D) superconductors are especially sensitive, with notable suppression due to enhanced localization and phase fluctuations (Refs. 14–19). Nonetheless, disorder can enhance superconductivity by increasing local pairing and multifractality of electronic states (Refs. 20, 24–27, 50), with empirical indications in quasi-1D MoSe chain crystals, TaSx monolayers, and NbSe2 monolayers (Refs. 21–23). Previous theoretical works have largely assumed uncorrelated disorder, though many disordered systems—cold atoms, photonic systems, and functional materials—exhibit long-range spatially correlated disorder that can qualitatively alter transport, mobility edges, and phase transitions (Refs. 32–47). Models with power-law correlations (“speckle”-like) have been used to study Anderson localization, mobility edges, and percolation (Refs. 57–63). Experimental analyses of dirty superconducting films reveal long-range disorder correlations (Ref. 23), motivating a direct microscopic study of their impact on superconductivity.

• Model and disorder: A 2D tight-binding BCS Hamiltonian with on-site attractive s-wave pairing is used on an N x N square lattice (nearest-neighbor hopping set to 1). Site disorder Vi includes long-range spatial correlations with Fourier-space structure factor SV(q) proportional to q^-α at small q (α controls correlation degree; α=0 is uncorrelated). Real-space disorder correlation length ξV is computed from the disorder correlator. Vi is shifted to zero mean; the disorder strength is defined by the variance of Vi.
• Microscopic equations: Self-consistent Hartree–Fock Bogoliubov–de Gennes (HF-BdG) equations are solved at T=0 including the Hartree potential Ui. The order parameter Δi and Ui are obtained from self-consistency. Long-range Coulomb repulsion is neglected; only s-wave pairing is considered.
• Observables and analyses: Spatial maps of Δi are computed versus disorder strength V and correlation α. Correlations between Δi and Vi are quantified by a statistical correlator C_AV. The distribution P(Δ) of the local gap magnitude is computed and fitted to a log-normal form at appropriate regimes; deviations are analyzed for large V and α. Spatial correlations of the order parameter are characterized by Sα(r,r') = ⟨Δα(r)Δα†(r')⟩ and its Fourier transform Sα(q) to extract a small-q power-law Sα(q) ∝ q^-β and a long-range correlation length ξξ from the second moment. The BCS coherence length ξ is extracted from the Cooper-pair correlation S(r,r') = ⟨|Δ(r,r')|^2⟩. Superfluid stiffness is calculated within BdG linear response: D0 = Axx(q→0,ω=0) + (−K), where Axx is paramagnetic current-current correlator and −K is the diamagnetic kinetic term. Phase fluctuations are incorporated via an effective XY model and self-consistent harmonic approximation to obtain the renormalized stiffness Ds = D0 exp(−⟨(δθ)^2⟩/2). Diamagnetic and paramagnetic contributions are analyzed separately. A phase diagram in the α–V plane is constructed from vanishing of stiffness (bare and fluctuation-corrected).
• Numerical setup: Lattice size N=40 with periodic boundary conditions; average filling n=0.875 (away from half filling). Coupling g=1 and large Debye window ωD→5 involve all single-particle states; typical average Δ ≈ 0.04, EF ≈ 3.76 (Δ/EF ≈ 0.01, weak coupling). Ensemble averages over Ns=50 disorder realizations for fixed V and α.

• Spatial profiles: Increasing disorder strength V induces inhomogeneous superconductivity with superconducting (S) clusters amid normal (N) regions. Increasing correlation degree α makes Δi more homogeneous, shrinks N clusters, and aligns cluster sizes with disorder textures, increasing correlation between Δ and V.
• Correlation between Δ and V: For α=0 and V=1, the correlator indicates a weak negative correlation (C_AV ≈ −0.5), consistent with superconductivity favoring potential minima. Correlation strengthens as α increases due to matched length scales (ξV comparable to cluster size).
• Order-parameter distribution: For uncorrelated disorder (α=0), P(Δ) is well fit by a log-normal distribution across V. As V increases, the peak shifts to smaller Δ and probability of near-zero Δ rises. With correlated disorder (α>0), the peak shifts to larger Δ and the distribution narrows (reduced σ) relative to the same V, reflecting enhanced robustness. At large V and α (e.g., α=3 at V=1.5; α≥2 at V=2), P(Δ) deviates from log-normal at small Δ, approaching a finite value as Δ→0 and exhibiting an opposite low-Δ slope.
• Quantitative distribution trends: Distribution width σ increases with V and decreases with α; the peak position Δmax decreases with V and increases with α. The probability P(Δ≈0) (Δ<0.1⟨Δ⟩) increases with V and decreases with α, indicating fewer normal regions at higher α.
• Long-range superconducting correlations: The order-parameter correlation Sα(r) retains a nonzero large-distance residue that grows with α. Its Fourier transform exhibits Sα(q) ∝ q^-β at small q; β is small for α≤1 but rises sharply for α≥2. The fit β ≈ β0 + (α/4)^γ with β0 ≈ 1.25 and γ ≈ 2.75 captures the trend, suggesting α≈2 marks a crossover between regimes.
• Correlation lengths: The BCS coherence length ξ and long-range correlation length ξξ both decrease with increasing V and increase with α. ξV (disorder correlation length) remains much smaller than ξ across parameters.
• Superfluid stiffness and phase diagram: The mean-field stiffness D0 and fluctuation-corrected Ds both decrease with V; phase fluctuations further suppress D. However, increasing α significantly enhances stiffness and shifts the critical disorder strength (where stiffness vanishes) to larger V. For α=0, stiffness vanishes already for V ≳ 0.75, despite a nonzero order parameter across most of the sample at V=1.0, illustrating that nonzero local Δ is insufficient for global phase coherence. Diamagnetic (−K) depends weakly on α and decreases slowly with V (∝ total density), whereas the paramagnetic Axx increases with disorder and depends on both V and α due to noncommutativity of current and Hamiltonian. The α–V phase diagram shows the superconducting (finite stiffness) region expands monotonically with α, with steeper boundary for α≥2.
• Overall: Long-range disorder correlations counteract the destructive effects of disorder on both local pairing and global phase coherence, rendering superconductivity more robust in 2D disordered systems at T=0.

The results demonstrate that introducing long-range correlations into the disorder potential substantially mitigates disorder-induced suppression of superconductivity. Correlations shift the balance between local pairing and global coherence: they smooth potential landscapes on a scale comparable to superconducting clusters, reduce the prevalence and size of normal regions, and enhance alignment between order-parameter clusters and low-potential textures. This yields narrower, higher-Δ order-parameter distributions and stronger long-range correlations (higher β). Importantly, the global superconducting response—captured by superfluid stiffness—increases with α for a fixed disorder strength, pushing the stiffness-loss threshold to higher V. Thus, the primary research question is answered affirmatively: spatial correlations in disorder make superconductivity more robust. The identification of α≈2 as a crossover in correlation scaling suggests distinct regimes of disorder structuring. These insights imply that not only impurity density but also spatial correlation properties are viable control knobs for engineering superconductivity in low-dimensional materials where correlated defects/dislocations are prevalent.

By solving self-consistent HF-BdG equations on a disordered 2D lattice at zero temperature with a tunable power-law-correlated disorder, the study shows that long-range correlations in the disorder potential enhance superconductivity locally and globally. Key contributions include: (i) quantification of how increasing α suppresses normal-region formation, narrows and shifts the Δ distribution to larger values, and strengthens long-range superconducting correlations; (ii) identification of a rapid increase of the correlation-spectrum exponent β for α≥2; (iii) demonstration that both the BCS coherence length and a longer superconducting correlation length grow with α and shrink with V; and (iv) establishment of an α–V phase diagram showing that superfluid stiffness—and thus global superconductivity—is significantly more robust for larger α. These findings elevate disorder correlations as a design parameter for tailoring superconducting properties. Future work could extend to finite temperatures and Tc, include long-range Coulomb interactions and unconventional pairing symmetries (e.g., d-wave), explore larger system sizes and dimensionalities, refine phase-fluctuation treatments beyond harmonic approximations, and interface with experiments to quantify disorder-correlation spectra in real materials.

• Zero-temperature analysis only; temperature dependence and Tc are not computed. - Mean-field HF-BdG framework with phase fluctuations treated in a self-consistent harmonic approximation; quantum fluctuations beyond this are not fully included. - Long-range Coulomb interactions are neglected; only on-site s-wave attraction is considered. - Specific disorder model with power-law correlations; other correlation types are not explored. - Finite lattice size (N=40) and a finite number of disorder realizations (Ns=50) may leave residual finite-size/statistical effects. - d-wave or other unconventional pairings are not treated (noted that correlated disorder might impact them differently). - Large Debye window and parameter choices simplify the pairing spectrum; material-specific details are not modeled.